Question

Find the perimeter of an equilateral triangle whose area is equal to that of a triangle with sides 21cm,16cm,and 13cm.Answer correct to 2 decimal places.

Answer 1

Given :

• Sides of the triangle :

⠀⠀⠀⠀⠀⠀⠀⠀⠀a = 21 cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀b = 16 cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀c = 13 cm

• Area of Equilateral triangle = Area of Scalene triangle.

To find :

The perimeter of the triangle.

Solution :

First let us find the area of the Scalene triangle.

Area of the Scalene triangle :

We know the formula for area of an Scalene triangle i.e,

$\underline{\bf{A = \sqrt{s(s - a)(s - b)(s - c)}}}$

Where :-

• a , b and c = Sides of the triangle
• s = Semi-perimeter

Here,

Semi-Perimeter = $\bf{s = \dfrac{a + b + c}{2}}$

So first let us find the semi-perimeter of the triangle .

Using the formula for Semi-perimeter and substituting the values in it, we get :

$:\implies \bf{s = \dfrac{a + b + c}{2}}$

$:\implies \bf{s = \dfrac{21 + 16 + 13}{2}}$

$:\implies \bf{s = \dfrac{50}{2}}$

$:\implies \bf{s = 25}$

$\boxed{\therefore \bf{Semi-Perimeter = 25\:cm}}$

Hence, the semi-perimeter of the triangle is 25 cm.

Now using the formula for area of a Scalene triangle and substituting the values in it, we get :

$:\implies \bf{A = \sqrt{s(s - a)(s - b)(s - c)}}$

$:\implies \bf{A = \sqrt{25(25 - 21)(25 - 16)(25 - 13)}}$

$:\implies \bf{A = \sqrt{25 \times 4 \times 9 \times 12}}$

$:\implies \bf{A = \sqrt{10800}}$

$:\implies \bf{A = 6\sqrt{3}}$

$\boxed{\therefore \bf{Area = 60\sqrt{3}}\:cm^{2}}$

Hence, the area of the Scalene triangle is 60√3 cm².

Side of the equilateral triangle :

We know :-

• Area of the equilateral triangle = Area of the Scalene triangle = 60√3 cm².

Using the formula for area of an equilateral triangle and substituting the values in it, we get :

$:\implies \bf{A = \dfrac{\sqrt{3} a^{2}}{4}}$

$:\implies \bf{60\sqrt{3} = \dfrac{\sqrt{3} a^{2}}{4}}$

$:\implies \bf{\dfrac{60\sqrt{3}}{\sqrt{3}} = \dfrac{a^{2}}{4}}$

$:\implies \bf{\dfrac{60\not{\sqrt{3}}}{\not{\sqrt{3}}} = \dfrac{a^{2}}{4}}$

$:\implies \bf{60 = \dfrac{a^{2}}{4}}$

$:\implies \bf{60 \times 4 = a^{2}}$

$:\implies \bf{240 = a^{2}}$

$:\implies \bf{\sqrt{240} = a^{2}}$

$:\implies \bf{15.49 = a}$

$\boxed{\therefore \bf{Side\:(a) = 15.49\:cm}}$

Hence the side of the triangle is 15.49 cm.

Perimeter of the triangle :

Using the formula for perimeter of a equilateral triangle and substituting the values in it, we get :

$:\implies \bf{P = 3 \times side}$

$:\implies \bf{P = 3 \times 15.49}$

$:\implies \bf{P = 46.47}$

$\boxed{\therefore \bf{Perimeter\:(P) = 46.47\:cm}}$

Hence, the perimeter of the equilateral triangle is 46.47 cm.